#### You may also like A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular? As a quadrilateral Q is deformed (keeping the edge lengths constnt) the diagonals and the angle X between them change. Prove that the area of Q is proportional to tanX. ### Multiplication of Vectors

An account of multiplication of vectors, both scalar products and vector products.

# Air Routes

##### Age 16 to 18 Challenge Level: London is situated at longitude $0^o$, latitude $52^o$ North and Cape Town at longitude $18^o$ East, latitude $34^o$ South. Taking the earth to be a sphere with unit radius (and ultimately scaling by 6367 kilometres for the radius of the earth) work out coordinates for both places, then find the angle LOC where L represents London, O the centre of the earth and C Cape Town. Hence find the distance on the surface of the earth between the two places. If a plane flies at an altitude of 6 kilometres and the journey takes 11 hours what is the average speed?

[You might also like to try the problems 'Over The Pole', which is a little simpler, and 'Flight Path' which is similar to 'Air Routes' but the method of solution given there is a bit different. ]